Rewrite the cosine function f(x)=cos(

Question

3
1
​
x−
6
π
​
) as a translated sine function.
A. f(x)=sin(
3
1
​
x+
3
π
​
)
B. f(x)=sin(
3
1
​
x+π)
C. f(x)=sin(
3
1
​
x)
D. f(x)=sin(
3
1
​
(x+

Answer 1

To rewrite the cosine function \( f(x) = \cos\left(\frac{3}{1}x - \frac{6\pi}{1}\right) \) as a translated sine function, we can use the identity:

\[ \cos(\theta) = \sin\left(\theta + \frac{\pi}{2}\right) \]

Using this identity, we can express \( f(x) \) in terms of sine:

1. Start with \( f(x) = \cos\left(3x - 6\pi\right) \).

2. We can rewrite this as: \[ f(x) = \sin\left(\left(3x - 6\pi\right) + \frac{\pi}{2}\right) \]

3. Simplifying the inside: \[ = \sin\left(3x - 6\pi + \frac{\pi}{2}\right) \] \[ = \sin\left(3x - \frac{12\pi}{2} + \frac{\pi}{2}\right) \] \[ = \sin\left(3x - \frac{11\pi}{2}\right) \]

4. Further simplifying \( -\frac{11\pi}{2} \) leads us to the sine function expressing the cosine with a shift.

The correct form of the function in terms of sine, considering necessary shifts, matches none of the provided options directly. Care must be taken to revisit the endpoint translations, but generally follows that:

• We assumed a sine function could yield a cosine function through simple transformations driven by the cosine identity or periodicity. Importantly though, the provided options:

A. \( f(x) = \sin\left(\frac{3}{1} x + \frac{3\pi}{1}\right) \)

B. \( f(x) = \sin\left(\frac{3}{1} x + \pi\right) \)

C. \( f(x) = \sin\left(\frac{3}{1} x\right) \)

D. \( f(x) = \sin\left(\frac{3}{1} (x + \ldots)\right) \)

None directly encapsulate the shifted transformation derived from \( f(x) = \sin\left(3x - \frac{11\pi}{2}\right) \). Revisit cosine's translated relationships for specific fitting options if needed.

Hence preferred consideration rests on the relativity of angular periodicity when determining true captures within the equation frameworks provided initially.